Generalizations of Choi's Orthogonal Latin Squares and Their Magic Squares Jon-Lark Kim Department of Mathematics Sogang University Seoul, 121-742, South Korea [email protected] ∗ Dong Eun Ohk Department of Mathematics Sogang University Seoul, 121-742, South Korea Doo Young Park Department of Mathematics Sogang University Seoul, 121-742, South Korea arXiv:1812.02202v2 [math.CO] 28 Feb 2020 Jae Woo Park Department of Mathematics Sogang University Seoul, 121-742, South Korea 2/28/2020 1 Abstract Choi Seok-Jeong studied Latin squares at least 60 years earlier than Euler ([3], [10]). He introduced a pair of orthogonal Latin squares of order 9 in his book [2]. Interestingly, his two orthogonal non-diagonal Latin squares produce a magic square of order 9, whose theoretical reason was not studied. There have been a few studies on Choi's Latin squares of order 9. The most recent one is Ko-Wei Lih's construction of Choi's Latin squares of order 9 based on two 3 × 3 orthogonal Latin squares [8]. In this paper, we give a new generalization of Choi's orthogonal Latin squares of order 9 to orthogonal Latin squares of size n2 using the Kronecker product including Lih's construction. We find a geometric description of Chois' orthogonal Latin squares of order 9 using the dihedral group D8. We also give a new way to construct magic squares from two orthogonal non-diagonal Latin square, which explains why Choi's Latin squares produce a magic square of order 9. Key Words: Choi Seok-Jeong, Koo-Soo-Ryak, Latin squares, magic squares AMS subject classification: 05B15, 05B20 1 Introduction A Latin square of order n is an n × n array in which n distinct symbols are arranged so that each symbol occurs once in each row and column. This Latin square is one of the most inter- esting mathematical objects. It can be applied to a lot of branches of discrete mathematics including finite geometry, coding theory and cryptography [4], [8]. In particular, orthogonal Latin squares have been one of main topics in Latin squares. The superimposed pair of two orthogonal Latin squares is also called a Graeco-Latin sqaure by Leonhard Euler (1707-1783) in 1776 [5]. It is known that the study of Latin squares was inceptively researched by Euler in 18th century. However the Korean mathematician, Choi Seok-Jeong [Choi is a family name] (1646-1715) already studied Latin squares at least 60 years before Euler's work. A pair of two Latin squares of order 9 was introduced in Koo-Soo-Ryak (or Gusuryak) written by Choi Seok-Jeong. The Koo-Soo-Ryak was listed as the first literature on Latin squares in the Handbook of Combinatorial Designs [3]. Let K be the matrix form of the superimposed Latin square of order 9 from Koo-Soo- Ryak: ∗Corresponding authors. J.-L. Kim was supported by Basic Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2013R1A1A2005172). 2 (5; 1) (6; 3) (4; 2) (8; 7) (9; 9) (7; 8) (2; 4) (3; 6) (1; 5) (4; 3) (5; 2) (6; 1) (7; 9) (8; 8) (9; 7) (1; 6) (2; 5) (3; 4) (6; 2) (4; 1) (5; 3) (9; 8) (7; 7) (8; 9) (3; 5) (1; 4) (2; 6) (2; 7) (3; 9) (1; 8) (5; 4) (6; 6) (4; 5) (8; 1) (9; 3) (7; 2) K = (1; 9) (2; 8) (3; 7) (4; 6) (5; 5) (6; 4) (7; 3) (8; 2) (9; 1) (3; 8) (1; 7) (2; 9) (6; 5) (4; 4) (5; 6) (9; 2) (7; 1) (8; 3) (8; 4) (9; 6) (7; 5) (2; 1) (3; 3) (1; 2) (5; 7) (6; 9) (4; 8) (7; 6) (8; 5) (9; 4) (1; 3) (2; 2) (3; 1) (4; 9) (5; 8) (6; 7) (9; 5) (7; 4) (8; 6) (3; 2) (1; 1) (2; 3) (6; 8) (4; 7) (5; 9) Then we can separate K into two Latin squares L and N. 5 6 4 8 9 7 2 3 1 1 3 2 7 9 8 4 6 5 4 5 6 7 8 9 1 2 3 3 2 1 9 8 7 6 5 4 6 4 5 9 7 8 3 1 2 2 1 3 8 7 9 5 4 6 2 3 1 5 6 4 8 9 7 7 9 8 4 6 5 1 3 2 L = 1 2 3 4 5 6 7 8 9 N = 9 8 7 6 5 4 3 2 1 3 1 2 6 4 5 9 7 8 8 7 9 5 4 6 2 1 3 8 9 7 2 3 1 5 6 4 4 6 5 1 3 2 7 9 8 7 8 9 1 2 3 4 5 6 6 5 4 3 2 1 9 8 7 9 7 8 3 1 2 6 4 5 5 4 6 2 1 3 8 7 9 To get a visible effect, let color in the squares. Figure 1: A colored two Latin squares L and N, respectively We paint colors for each numbers, 1; 2; ··· ; 9. In detail, 1; 2; 3 are colored in red, 4; 5; 6 are colored in green, and 7; 8; 9 are colored in blue. Then we observe that the Latin squares have self-repeating patterns. This simple structure of Choi's Latin squares motivates some 3 generalization of his idea. We generalize Choi's Latin squares in three directions: the Kro- necker product approach, the Dihedral group approach, and magic squares from Choi's Latin squares. In this paper, we give a new generalization of Choi's orthogonal Latin squares of order 9 to orthogonal Latin squares of size n2 using the Kronecker product including Lih's construction. We find a geometric description of Chois' orthogonal Latin squares of order 9 using the dihedral group D8. We also give a new way to construct magic squares from two orthogonal non-diagonal Latin square, which explains why Choi's Latin squares produce a magic square of order 9. 2 A generalization of Choi's orthogonal Latin squares Definition 1. A Latin square of order n is an n × n array (or matrix) in which n distinct symbols are arranged so that each symbol occurs once in each row and column. For example, 1 2 3 2 3 1 3 1 2 is a Latin square of order 3. In this article, we assume that a Latin square of order n has the entries 1; 2; ···; n instead of n symbols. Definition 2. Let A and B be Latin squares of order n. We said that the two Latin squares A and B are orthogonal if each ordered pair of its entry occurs exactly once when A is superimposed on B. For example, if the two Latin squares A and B are given by 1 2 3 3 2 1 A = 2 3 1 ;B = 1 3 2 3 1 2 2 1 3 then the superimposed square is (1; 3) (2; 2) (3; 1) (2; 1) (3; 3) (1; 2) (3; 2) (1; 1) (2; 3) because (i; j)(i; j 2 f1; 2; 3g) occurs only once. Thus A and B are orthogonal Latin squares. 4 Definition 3. ([8]) Let A = (aij) be a Latin square of order n(i; j 2 f1; 2; ··· ; ng) and B = (bst) be a Latin square of order m(s; t 2 f1; 2; ··· ; mg). Then the Kronecker product of A and B, which is an mn × mn square A ⊗ B given by (a11;B)(a12;B) ··· (a1n;B) (a21;B)(a22;B) ··· (a2n;B) A ⊗ B = . .. (an1;B)(an2;B) ··· (ann;B) where (aij;B) is the m × m square (aij; b11)(aij; b12) ··· (aij; b1m) (aij; b21)(aij; b22) ··· (aij; b2m) (aij;B) = . .. (aij; bm1)(aij; bm2) ··· (aij; bmm) Lemma 2.1. ([8]) A ⊗ B is a Latin square if A and B are both Latin squares. Theorem 2.2. ([8]) If two Latin squares A1 and A2 of order n are orthogonal and two Latin squares B1 and B2 of order m are orthogonal, then A1 ⊗ B1 and A2 ⊗ B2 of order mn are orthogonal. Now it is natural to substitute m(aij − 1) + bkl for the entry (aij; bkl) in A ⊗ B. Thus we define the substituted Kronecker product ⊗S of two Latin squares A and B by the following block matrix 2 3 (m(a11 − 1) × Nm + B) ··· (m(a1n − 1) × Nm + B) 6 . .. 7 A ⊗S B = 4 . 5 (m(an1 − 1) × Nm + B) ··· (m(ann − 1) × Nm + B) where A = (aij) is a matrix of order n, B is a matrix of order m and Nm is the m × m all one matrix. Let us return to the Latin squares. Judging from Figure 2, we can expect that L is closely related to a Latin square of order 3. In fact, let A3 = (aij); (i; j 2 f1; 2; 3g) by 2 3 1 A3 = 1 2 3 3 1 2 Then the following block matrix 2 3 (3(a11 − 1) × N3 + A3) (3(a12 − 1) × N3 + A3) (3(a13 − 1) × N3 + A3) 6 7 4(3(a21 − 1) × N3 + A3) (3(a22 − 1) × N3 + A3) (3(a23 − 1) × N3 + A3)5 (3(a31 − 1) × N3 + A3) (3(a32 − 1) × N3 + A3) (3(a33 − 1) × N3 + A3) 5 produces L. In other words, L = A3 ⊗S A3. Similarly, if we let B3 = (bij); (i; j 2 f1; 2; 3g) by 1 3 2 B3 = 3 2 1 2 1 3 then the following block matrix 2 3 (3(b11 − 1) × N3 + B3) (3(b12 − 1) × N3 + B3) (3(b13 − 1) × N3 + B3) 6 7 4(3(b21 − 1) × N3 + B3) (3(b22 − 1) × N3 + B3) (3(b23 − 1) × N3 + B3)5 (3(b31 − 1) × N3 + B3) (3(b32 − 1) × N3 + B3) (3(b33 − 1) × N3 + B3) produces N.